X₂₂₁

X₁-CEVA CONJUGATE OF X₅₆

X₂₂₁ = crosssum of X₁ and X₈₄

5, 6, 7 3, 4, 5 Kepler Kimberling

X₂₂₁ (X₁-CEVA CONJUGATE OF X₅₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 19}, {1, 84}

• Related Centers: X₁ (INCENTER), X₅₆ (EXSIMILICENTER), X₆ (SYMMEDIAN POINT), X₁₉ (CLAWSON POINT), X₈₄ (ISOGONAL CONJUGATE OF X₄₀)

Trilinears: \((\sin^{2} A/2)(\cos B + \cos C - \cos A - 1) : :\)

Search Value (6, 9, 13): \(-0.330449218124\)

▸ Alternate Coordinates1 form

• Trilinears: \((1 - \cos A)(1 + \cos A - \cos B - \cos C) : :\)

▸ Incident Central Lines10 lines

{1, 84}, {3, 102}, {6, 19}, {8, 651}, {31, 56}, {40, 223}, {55, 64}, {201, 220}, {204, 207}, {960, 1038}

▸ Functional Relations9 relations

• X₂₂₁ = isogonal conjugate of X₂₈₀

• X₂₂₁ = anticomplement of X₂₀₃₀₆

• X₂₂₁ = X_i-Ceva conjugate of X_j for these (i,j): (1, 56), (222, 6), (223, 198)

• X₂₂₁ = crosspoint of X_i and X_j for these (i,j): (1, 40), (196, 347)

• X₂₂₁ = crosssum of X₁ and X₈₄

• X₂₂₁ = X_i-beth conjugate of X_j for these (i,j): (1, 34), (40, 40), (101, 220), (109, 221), (110, 3)

• X₂₂₁ = trilinear product X₄₀*X₅₆

• X₂₂₁ = X₉₂-isoconjugate of X(268

• X₂₂₁ = perspector of unary cofactor triangles of 1st and 3rd extouch triangles

Notes

Let I = X₁ = incenter, and A'B'C' = medial triangle. Let A_B = AC∩IA', and define B_C and C_A cyclically. Let A_C = AB∩IA', and define B_A and C_B cyclically. Let O_A be the circumcircle of AA_BA_C, and define O_B and O_C cyclically. Then X₂₂₁ is the radical center of O_A, O_B, O_C. (Angel Montesdeoca, April 27, 2021)